Epigroups in which the relation of having the same pseudo-inverse is a congruence. (Q372350)

An epigroup is a semigroup in which some power of each element belongs to a subgroup. After sporadic early work, a comprehensive study of such semigroups was made by \textit{L. N. Shevrin} [Russ. Acad. Sci., Sb., Math. 82, No. 2 485-512 (1995); translation from Mat. Sb. 185, No. 7, 129-160 (1994; Zbl 0839.20073)]. Generalizing the universal algebraic approach to completely regular semigroups, epigroups form a variety when equipped with the unary operation \(a\mapsto\overline a\), where if \(a^\omega\) is the identity element of the subgroup that contains a power of \(a\), then \(\overline a\) -- the pseudo-inverse of \(a\) -- is the inverse of \(a^\omega a\) in that subgroup. The main result of the paper provides a number of characterizations of those epigroups in which the equivalence relation in the title is a congruence. For instance this class is the Malcev product of the variety of nilsemigroups with the variety of completely regular semigroups; alternative characterizations are by means of forbidden divisors and by means of identities. Further results include characterizations of various subvarieties in similar terms.